dist.Multivariate.t.Cholesky {LaplacesDemon} | R Documentation |

## Multivariate t Distribution: Cholesky Parameterization

### Description

These functions provide the density and random number generation for the multivariate t distribution, otherwise called the multivariate Student distribution, given the Cholesky parameterization.

### Usage

```
dmvtc(x, mu, U, df=Inf, log=FALSE)
rmvtc(n=1, mu, U, df=Inf)
```

### Arguments

`x` |
This is either a vector of length |

`n` |
This is the number of random draws. |

`mu` |
This is a numeric vector or matrix representing the location
parameter, |

`U` |
This is the |

`df` |
This is the degrees of freedom, and is often represented
with |

`log` |
Logical. If |

### Details

Application: Continuous Multivariate

Density:

`p(\theta) = \frac{\Gamma[(\nu+k)/2]}{\Gamma(\nu/2)\nu^{k/2}\pi^{k/2}|\Sigma|^{1/2}[1 + (1/\nu)(\theta-\mu)^{\mathrm{T}} \Sigma^{-1} (\theta-\mu)]^{(\nu+k)/2}}`

Inventor: Unknown (to me, anyway)

Notation 1:

`\theta \sim \mathrm{t}_k(\mu, \Sigma, \nu)`

Notation 2:

`p(\theta) = \mathrm{t}_k(\theta | \mu, \Sigma, \nu)`

Parameter 1: location vector

`\mu`

Parameter 2: positive-definite

`k \times k`

scale matrix`\Sigma`

Parameter 3: degrees of freedom

`\nu > 0`

(df in the functions)Mean:

`E(\theta) = \mu`

, for`\nu > 1`

, otherwise undefinedVariance:

`var(\theta) = \frac{\nu}{\nu - 2} \Sigma`

, for`\nu > 2`

Mode:

`mode(\theta) = \mu`

The multivariate t distribution, also called the multivariate Student or
multivariate Student t distribution, is a multidimensional extension of the
one-dimensional or univariate Student t distribution. A random vector is
considered to be multivariate t-distributed if every linear
combination of its components has a univariate Student t-distribution.
This distribution has a mean parameter vector `\mu`

of length
`k`

, and an upper-triangular `k \times k`

matrix that is
Cholesky factor `\textbf{U}`

, as per the `chol`

function for Cholesky decomposition. When degrees of freedom
`\nu=1`

, this is the multivariate Cauchy distribution.

In practice, `\textbf{U}`

is fully unconstrained for proposals
when its diagonal is log-transformed. The diagonal is exponentiated
after a proposal and before other calculations. Overall, the Cholesky
parameterization is faster than the traditional parameterization.
Compared with `dmvt`

, `dmvtc`

must additionally
matrix-multiply the Cholesky back to the scale matrix, but it
does not have to check for or correct the scale matrix to
positive-definiteness, which overall is slower. The same is true when
comparing `rmvt`

and `rmvtc`

.

### Value

`dmvtc`

gives the density and
`rmvtc`

generates random deviates.

### Author(s)

Statisticat, LLC. software@bayesian-inference.com

### See Also

`chol`

,
`dinvwishartc`

,
`dmvc`

,
`dmvcp`

,
`dmvtp`

,
`dst`

,
`dstp`

, and
`dt`

.

### Examples

```
library(LaplacesDemon)
x <- seq(-2,4,length=21)
y <- 2*x+10
z <- x+cos(y)
mu <- c(1,12,2)
S <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
U <- chol(S)
df <- 4
f <- dmvtc(cbind(x,y,z), mu, U, df)
X <- rmvtc(1000, c(0,1,2), U, 5)
joint.density.plot(X[,1], X[,2], color=TRUE)
```

*LaplacesDemon*version 16.1.6 Index]